I have Safari 10.12.6 and I get messages that my browser isn't up to date. Is there no longer updates for Safari on certain Macs. I have a MacBook Pro from 2016

Safari updates when you update macOS.

The model year of a MacBook Pro is available at Apple menu > About This Mac.

A late-2013 or later can be upgraded to the current version of Big Sur. See: How to upgrade to macOS Big Sur - Apple Support

Hello. It sounds like you are running an old version of macOS which is no longer receiving Safari updates. If you want your browser to be up-to-date again, it is recommended to move to a newer version of macOS. The good news is that your Mac is compatible with macOS Big Sur, so you should be able to update to the latest version of macOS (Big Sur) by following the steps here.

I think I can explain what is happening. If you are unable to access certain web sites and possibly getting a message that your browser must be updated, this is not an unusual situation. The reason is that web site engineers write their software to work with various browsers based on standardized specifications if they are to be compliant. Some sites are not yet compliant. Also, some browsers may not be fully compliant. You can hope that your web site's developers eventually update their site to work with Safari, you can upgrade Safari or you can use another browser that does work such as Firefox or Chrome. They can be individually updated as needed. It is important to know that Safari is installed as part of the operating system and it normally it cannot be individually updated. Safari is periodically updated when the computer's operating system (OS) is updated or upgraded. Finally, even if you are able to update Safari, there is no guarantee that it will be compatible with every web site. This is why many of us have more than one browser installed.